## Properties and Applications of Hawkes Processes

##### Summary

Point processes in probability theory are special stochastic processes which model the occurrence
of events (points) in some space and parameterized by their frequency. Hawkes
processes are general point processes where the frequency can be self-exciting or mutually
exciting. They occur in various areas such as geological sciences, epidemiology and financial
mathematics. The focus of this thesis lies in discussing different types of Hawkes processes
and their properties as well as presenting financial applications. The (linear) Hawkes process
or self-exciting Hawkes process is a process with a single counting process such that that the
occurrence of an event increases the probability of the occurrence of another event. In the
case of the (linear) Hawkes process, we prove the Law of Large Numbers and the Central
Limit Theorem. Moreover, we study the Hawkes likelihood function and the Hawkes loglikelihood
function. Besides a self-exciting Hawkes process there exists a mutually exciting
Hawkes process, which is a process with multiple counting processes that are depended on
each other. Meaning that the occurrence of an event in one of these counting processes
also lead to an increased probability of an event occurring in the other counting processes.
We study the Hawkes likelihood function and the Hawkes log-likelihood function for the
mutually exciting Hawkes process. The marked Hawkes process is a (linear) Hawkes process
with added random variables called the random marks. We prove the Hawkes likelihood
function and the Hawkes log-likelihood function as well as the Central Limit Theorem.
Furthermore, we derive the dynamics for the Hawkes jump-diffusion model given by three
stochastic differential equations and prove the Law of Large Numbers and Central Limit
Theorem for this particular model.